Given a directed graph $G = (V, E)$ with all edge weights being non-negative and two disjoint subsets of nodes $S, T \subseteq V$, design an algorithm to find the shortest paths among $S$ and $T$, (formally, $\min\{d(s,t): s \in S, t \in T\}$) in $O(m \log n)$ time, where $|V| = n$ and $|E| = m$.

Note: The first version of my problem is to find all $d(s,t)$ for each $s \in S$ and $t \in T$. As @D.W. argues, it is not likely to do this in $O(m \log n)$ time because taking $S, T = V$ requires all pairs shortest paths.

Given a directed graph $G = (V, E)$ with all edge weights being non-negative and two disjoint subsets of nodes $S, T \subseteq V$, design an algorithm to find the shortest paths among $S$ and $T$, (formally, $\min\{d(s,t): s \in S, t \in T\}$) in $O(m \log n)$ time, where $|V| = n$ and $|E| = m$.

Given a directed graph $G = (V, E)$ with all edge weights being non-negative and two disjoint subsets of nodes $S, T \subseteq V$, design an algorithm to find the shortest paths between any node in $S$ and any node in $T$ in $O(m \log n)$ time, where $|V| = n$ and $|E| = m$.

A first attempt is to add two extra nodes $s$ and $t$, connect them to each node in $S$ and $T$ respectively, and to find the shortest path from $s$ to $t$. However, in my opinion, this find only a shortest paths for each node in $T$, instead of all pairs shortest path between $S$ and $T$.

Given a directed graph $G = (V, E)$ with all edge weights being non-negative and two disjoint subsets of nodes $S, T \subseteq V$, design an algorithm to find the shortest paths among $S$ and $T$, (formally, $\min\{d(s,t): s \in S, t \in T\}$) in $O(m \log n)$ time, where $|V| = n$ and $|E| = m$.

Note: The first version of my problem is to find all $d(s,t)$ for each $s \in S$ and $t \in T$. As @D.W. argues, it is not likely to do this in $O(m \log n)$ time because taking $S, T = V$ requires all pairs shortest paths.